Fractions

Fractions describe parts of a whole. In Key Stage 1, children are introduced to halves and quarters in Year 1, and extend to thirds and more complex fractions in Year 2.

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What This Topic Covers

• Year 1: Recognise, find and name a half (1/2) and a quarter (1/4) of an object, shape or quantity
• Year 2: Recognise and write 1/3, 1/4, 2/4 and 3/4 of lengths, shapes, sets of objects and quantities; understand that 1/2 = 2/4

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Year 1: Halves and Quarters

What is a Fraction?

A fraction describes an equal part of a whole. The key word is equal — the parts must be the same size.

A half (1/2):

+-----+-----+
|XXXXX|     |
|XXXXX|     |
+-----+-----+
1 out of 2 equal parts

A quarter (1/4):

+--+--+--+--+
|XX|  |  |  |
|XX|  |  |  |
+--+--+--+--+
1 out of 4 equal parts

A Half (1/2)

A half is 1 of 2 equal parts. The whole is divided into 2 equal pieces, and a half is one of those pieces.

Halves of shapes:

• Half of a square is one rectangle
• Half of a circle is a semicircle

Halves of quantities:

• Half of 8 apples = 4 apples
• Half of 10 = 5
• Half of 20 = 10

Halves of lengths:

• Half of a 10 cm ribbon = 5 cm

A Quarter (1/4)

A quarter is 1 of 4 equal parts. The whole is divided into 4 equal pieces, and a quarter is one of those pieces.

Quarters of quantities:

• Quarter of 8 = 2
• Quarter of 12 = 3
• Quarter of 20 = 5

Connection: Finding a quarter is the same as halving, and then halving again. Quarter of 20: half of 20 = 10, then half of 10 = 5

flowchart TD
  Whole[1 whole]
  Whole --> H1[1/2]
  Whole --> H2[1/2]
  H1 --> Q1[1/4]
  H1 --> Q2[1/4]
  H2 --> Q3[1/4]
  H2 --> Q4[1/4]

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Year 2: Extending Fractions

The Language of Fractions

Every fraction has two parts:

Part	Name	What it means
Top number	Numerator	How many equal parts you have
Bottom number	Denominator	How many equal parts the whole is divided into

Example: 3/4 means numerator = 3, denominator = 4, so "3 out of 4 equal parts"

The Key Year 2 Fractions

Fraction	Name	What it means
1/2	one half	1 out of 2 equal parts
1/3	one third	1 out of 3 equal parts
1/4	one quarter	1 out of 4 equal parts
2/4	two quarters	2 out of 4 equal parts
3/4	three quarters	3 out of 4 equal parts

Fractions of Quantities

Fraction	Of 12	Of 24
1/2	6	12
1/3	4	8
1/4	3	6
3/4	9	18

How to find a fraction of a quantity:

1. Divide by the denominator (bottom number): 3/4 of 12 means 12 / 4 = 3
2. Multiply by the numerator (top number): 3 x 3 = 9

Worked example with more scaffolding: Find 3/4 of 20

Step 1 — What does the denominator (4) tell us? Divide into 4 equal groups. 20 ÷ 4 = 5 (each group has 5) Step 2 — What does the numerator (3) tell us? We need 3 of those groups. 5 × 3 = 15 So 3/4 of 20 = 15.

Check: 1/4 of 20 = 5, so 3/4 of 20 = 5 + 5 + 5 = 15 ✓

Non-Unit Fractions of Shapes

A unit fraction has a numerator of 1 (like 1/2, 1/3, 1/4). A non-unit fraction has a numerator greater than 1 (like 2/3, 3/4).

Shading 3/4 of a rectangle:

+--+--+--+--+
|XX|XX|XX|  |
|XX|XX|XX|  |
+--+--+--+--+
3 out of 4 equal parts shaded = 3/4

Shading 2/3 of a rectangle:

+----+----+----+
|XXXX|XXXX|    |
|XXXX|XXXX|    |
+----+----+----+
2 out of 3 equal parts shaded = 2/3

Remember: the parts must always be equal in size.

Equivalence: 1/2 = 2/4

This is the first fraction equivalence children learn. Two quarters make the same amount as one half:

+-----+-----+         +--+--+--+--+
|XXXXX|     |         |XX|XX|  |  |
+-----+-----+         +--+--+--+--+
    1/2                   2/4

1/2 = 2/4

Counting in Fractions

Children count along a number line in fractions. Each step is the same size — one quarter.

Counting in quarters from 0 to 2:

0        1/4      2/4      3/4       1       1 1/4    1 2/4    1 3/4      2
|---------|--------|--------|---------|---------|--------|--------|---------|
start   +1/4     +1/4     +1/4      +1/4     +1/4     +1/4     +1/4     +1/4

Each jump is +1/4. Notice:

• 2/4 is the same position as 1/2 (equivalent fractions!)
• After 3/4 comes 4/4, which equals 1 whole.
• After 1 3/4 comes 1 4/4, which equals 2 wholes.

Counting in thirds from 0 to 2:

0       1/3      2/3       1       1 1/3    1 2/3      2
|--------|--------|---------|--------|--------|---------|
start   +1/3    +1/3      +1/3    +1/3     +1/3      +1/3

Each jump is +1/3. After 2/3 comes 3/3, which equals 1 whole.

This reinforces that fractions are numbers on the number line, not just parts of shapes.

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Key Vocabulary

Term	Meaning
fraction	a part of a whole
numerator	the top number — how many parts you have
denominator	the bottom number — how many equal parts the whole is divided into
equal parts	parts that are exactly the same size
half (1/2)	one of two equal parts
quarter (1/4)	one of four equal parts
third (1/3)	one of three equal parts
three quarters (3/4)	three of four equal parts
equivalent	equal in value (e.g. 1/2 = 2/4)
whole	the complete, unbroken amount

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Common Misconceptions

• "Fractions do not have to be equal parts" — Many children split a shape into two parts of different sizes and call each a "half". Emphasise that both parts must be equal.
• "A bigger denominator means a bigger fraction" — 1/2 is greater than 1/4, even though 4 > 2. The more pieces you cut a whole into, the smaller each piece is.
• "Fractions are only for shapes" — Fractions apply to objects, quantities, lengths and numbers too.

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Worked Example for Parents and Teachers

Fractions in KS1 must be grounded in fair sharing. The single most useful approach is to teach every fraction concept through three representations: physical objects, pictures, and number lines — what mastery teaching calls Concrete-Pictorial-Abstract (CPA).

Imagine a Year 2 lesson with the objective: "Children will find 3/4 of a quantity."

Step 1 — Concrete. Place 12 counters on the table. Say, "We are going to find three-quarters of these counters. The bottom number — the denominator — tells us how many equal groups to make. The bottom number is 4. So we make 4 equal groups." Sort the 12 counters into 4 hoops (or piles) with 3 in each. Now ask, "How many counters in 1 group? Three. So 1/4 of 12 is 3."

Continue: "The top number — the numerator — tells us how many of those groups we want. We want 3 of them. Let me push 3 of the groups together." Push 3 piles together — there are 9 counters. "So 3/4 of 12 is 9."

Step 2 — Pictorial. On the board, draw the same example with circles in 4 hoops. Highlight 3 of the 4 hoops. The picture mirrors the concrete action exactly. Children draw their own version on whiteboards.

Step 3 — Abstract. Write the calculation: 3/4 of 12 = (12 ÷ 4) × 3 = 3 × 3 = 9. Read it aloud as the steps: divide by the denominator, multiply by the numerator. Repeat with new examples — 1/3 of 9, 2/4 of 8, 3/4 of 20 — always returning to the language equal groups.

Verbal prompts to use:

• "What does the bottom number tell us?"
• "What does the top number tell us?"
• "How many in each equal group?"
• "How do you know your answer is correct?"